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lecture4 - NETWORK PROBLEMS 1.224J/ESD.204J TRANSPORTATION...

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NETWORK PROBLEMS 1.224J/ESD.204J TRANSPORTATION OPERATIONS, PLANNING AND CONTROL: CARRIER SYSTEMS Professor Cynthia Barnhart Professor Nigel H.M. Wilson Fall 2003
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1/2/2004 Barnhart - 1.224J 2 Announcements –C. Barnhart open office hours from 1:00-2:30 on Wednesday this week –Reference: Network Flows: Theory, Algorithms, and Applications (Ahuja, Magnanti, Orlin)
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1/2/2004 Barnhart - 1.224J 3 Outline • Network Introduction • Properties of Network Problems • Minimum Cost Flow Problem • Shortest Path • Maximum Flow Problem • Assignment Problem • Transportation Problem • Circulation Problem • Multicommodity Flow problem • Matching Problem
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1/2/2004 Barnhart - 1.224J 4 Networks • Very common in transportation • Can be physical networks: transportation networks such as railroads and highways) • Network flows sometimes also arise in surprising ways (problems that on the surface might not appear to involve any networks at all). • Sometimes the nodes and arcs have a temporal dimension that models activities that take place over time. Many scheduling applications have this flavor (crew scheduling; location and layout theory; warehousing and distribution; production planning and control)
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1/2/2004 Barnhart - 1.224J 5 Graphs & Networks • Network problems are defined on graphs – Undirected and directed graphs G=(N, A) • N=set of nodes • A=set of feasible links/ arcs – Trees (connected graph; no cycles) – Bipartite graphs: two sets of nodes with arcs that join only nodes between the 2 sets Additional numerical information such as: b(i) representing supply and demand at each node i ; u ij representing the capacity of each arc ij l ij representing the lower bound on flow for each arc ij c ij representing the cost of each arc ij . 5 1 3 4 2 Undirected Graph 5 1 3 4 2 Directed Graph b(4) c 12 , u 12 b(2) b(1) c 21 , u 21 c 25 , u 25 c 54 , u 54 c 45 , u 45 c 35 , u 35 c 12 , u 12 b(3) b(5) 5 1 3 4 2 Network
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1/2/2004 Barnhart - 1.224J 6 Formulating network flow problems Concise formulation Node arc incidence matrix: Columns = arcs Rows = nodes Outgoing arc +1 Incoming arc -1 No arc incident 0 Sum of rows of A = 0 (1) Balance constraints: flow out minus flow in must equal the supply/demand at the node (2) Flow lower bound constraints (usually lower bound is not stated and equal to 0) (3) Capacity constraints (4) Integrality constraints {} ) 4 ....( .......... .......... .......... ) , ( , ) 3 ....( .......... .......... .......... ) , ( , ) 2 ..... ( .......... .......... .......... ) , ( , ) 1 ..( .......... ), ( . ) , ( : ) , ( : ) , ( A j i Z X A j i U X A j i L X N i i b X X t s X C Minimize ij ij ij ij ij A i j j ij A j i j ij A j i ij ij = +
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1/2/2004 Barnhart - 1.224J 7 Properties of Network Problems • Solving the LP relaxation of network problems with integer problem data, yields an integer solution • Network problems are special cases of LPs and any algorithm for a LP can be directly applied.
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lecture4 - NETWORK PROBLEMS 1.224J/ESD.204J TRANSPORTATION...

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